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The last couple of years symbolic computer algebra packages, such as Mathematica, Maple, and Matlab, have developed into a very user-friendly and high-level prototyping environment. Especially Mathematica combines the advantages of symbolic manipulation and processing with an advanced front-end text prcessor. This paper has been completely written in Mathematica version 4 as a notebook. The advantage is that this paper can be read as an interactive paper: the high-level code of any function is directly visible, and cn be executed immediately, as well as modified or templated for own use. Students can now use the exact code rather than pseudocode. With these high-level programming tools most programs can be expressed in very few lines, so it keeps the reader at a hghly intuitive but practical level. Mathematica notebooks are portable, and run on any system equivalently. ... The main focus of this paper is twofold: to provide a rehearsal of the derivation of the Gaussian kernel and its derivatives as an essential class of front-end vision aperture functions, and to provide a practical tutorial for a broad aucience to be able to do geometric reasoning with robust multiscale differential opertors on discrete images. This may break ground for the view of the front-end visual system as a geometry-engine, or "inference machine," rather than do a spatial frequency analysis. This paper can only focus on a small area, the differential geometry and its features. Much research is currently underway. An important area is especially the deep structure of images, where the relations between scales are studied.
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